(4) Reduce 7 oz. 4dwts. to the decimal of 1 lb., and 2qrs. 3nls. to the decimal of an English ell of five quarters. Answers: .6, and .555 &c. (5) Reduce 12hrs. 55 min. 23 sec. to the decimal of a day, and 5 days. 12 hrs. 25 min. 37.92 sec. to the decimal. of a week. Answers: .538461 &c., and . 788257 &c. (6) Express 12s. 6d., 15s. 9åd. and £4. 13s. 4 d. as decimals of £1. Answers: .628125, .790625, and 4.66875. (7) Reduce 1.1s. to the decimal of 10s., and 5s. to to the decimal of 13s. 4d. Answers: .11 and .375. (8) Find the values of .45 of £1., .16875 of £1. and 2.36875 of £1. 'Answers: 9s., 3s. 4 d., and £2. 7s. 44d. (9) Required the values of £.5675, .375 cwt., .6875 yds. and 13.3375 acres. Answers: 11s. 4}d., 1 qr. 14lbs., 2 qrs. 3 na., and 13 ac. 1 ro. 14 po. (10) What are the values of .203125 qrs., and .73625 bush.? Answers: 1bush. 2 pks. 1 gal., and 2 pks. 1 gal. 3qts. (11) What are the values of .07 of £2. 10s., and of .0474609375 of £10. 13s. 4d.? Answers: 3s. 6d., and 10s. 1 d. (12) Find the value of .5 shillings + .7 crowns + .125 pounds. Answer: 6s. 6d. (13) Reduce £24. 16s. 41d. and £167. 10s. 61d., to decimals of the same denomination; and find how often the former is contained in the latter. 108. RECURRING DECIMALS. DEF. In the conversion of a vulgar fraction into a decimal, if the division performed according to the rule laid down in article (103), do not terminate, but the figures of the quotient continually recur in some certain order, the result is called a recurring or circulating decimal: the quantity repeated is styled its period, and is frequently termed a simple or compound repetend, according as it consists of one or more figures: and the extent of the period is denoted by means of single points or dots placed over the first and last of the figures which compose it. If the quotient comprise other figures besides those which are repeated, it is called a mixed circulating decimal, consisting of a non-recurring and a recurring part. Ex. 1. Convert and into decimals. Proceeding according to the rule, we have the former having the simple repetend 3, and the latter the compound repetend 148, which being denoted by 3 and 148 respectively, 5 whence is equivalent to the mixed circulating decimal 36 .13888 &c., the non-recurring part being 13 and the recurring part 8, and the result is written 5 36 = .138. Conversely, every pure or mixed circulating decimal must be equal to, and expressible by, a vulgar fraction. 109. To find the vulgar fraction which shall be equivalent to a pure recurring decimal. Let the circulates be .666 &c., and .9696 &c., or .& and .96: then if, for the sake of conciseness, we suppose the symbols x and y to represent their values, we shall have the following operations: x= .666 &c. 10x = 6.666 &c. y = .9696 &c. 100 y = 96.9696 &c. whence subtracting in each case, the former from the latter, we obtain These results may be easily verified, and from them we derive the following rule. RULE. Make the repetend the numerator of a fraction whose denominator shall consist of as many nines as there are figures in the said repetend, and this reduced to its simplest terms will be the vulgar fraction required. 110. To find the vulgar fraction which shall represent the value of a mixed recurring decimal. Ex. To ascertain the vulgar fractions equivalent to .27 and .2457, we have whence, subtracting the second line from the third in each case, we find furnish us with a general rule. RULE. Make the non-recurring and the recurring parts taken together, diminished by the non-recurring part taken alone, the numerator of a fraction whose denominator shall consist of as many nines as there are recurring figures, followed by as many ciphers as there are non-recurring figures, and this reduced to its lowest terms will be the vulgar fraction required. 111. It will hence appear that the arithmetical operations upon recurring decimals, may be correctly effected by means of the same operations performed upon their equivalent vulgar fractions. Ex. Let it be required to find the sum, difference, product and quotient, of the recurring decimals .6 and .296. 8 1000x = 296.296 x=.296 27 : 8982=286 2 and .296 8 26 + .962: 3 27 27 2=296 999 -37×8 .370: 37x27 8 16 = .197530864: 27 2.25: 4 3 8 27 8 02100 2100 2100 = 3 27 = 916 the first three of which are recurring decimals, and the last a finite quantity when expressed decimally: and it may be remarked that the same results could have been obtained by the immediate operations only by means of a laborious process. 112. In the same manner recurring decimals of specified units may be treated, and their exact values thence obtained. Ex. Find the value of .16 of a pound sterling. 113. Since, in converting a vulgar fraction into a decimal, either 10, 100, 1000, &c., or their multiples, are divided by the denominator, it is evident that the decimal will terminate or not, according as these numbers are divisible by the denominator or not: whence, as the only incomposite factors of 10, 100, 1000, &c., are 2 and 5, it follows that vulgar fractions, whose denominators can be resolved into these factors, are equivalent to finite decimals, whilst all others are not. (1) What are the recurring decimals corresponding to the vulgar fractions, Answers: .307692, .12195 and .i509433962264. (3) Find the vulgar fractions equivalent to the recurring decimals: .5, .027 and .534. (4) What vulgar fractions will represent the values of the recurring decimals, .3621, .47543 and .6761904 ? |